The mapping account

The classic account of computation is found throughout the works of Hilary Putnam and others. Peter Godfrey-Smith has dubbed this the "simple mapping account."[3] Gualtiero Piccinini's summary of this account states that a physical system can be said to perform a specific computation when there is a mapping between the state of that system to the computation such that the “microphysical states [of the system] mirror the state transitions between the computational states.”[4]

The semantic account

Philosophers such as Jerry Fodor[5] have suggested various accounts of computation with the restriction that semantic content be a necessary condition for computation (that is, what differentiates an arbitrary physical system from a computing system is that the operands of the computation represent something). This notion attempts to prevent the logical abstraction of the mapping account of pancomputationalism, the idea that everything can be said to be computing everything.

The mechanistic account

Gualtiero Piccinini proposes an account of computation based in mechanical philosophy. It states that physical computing systems are types of mechanisms that, by design, perform physical computation, or “the manipulation (by a functional mechanism) of a medium-independent vehicle according to a rule.” Medium-independence requires that the property is able to be instantiated by multiple realizers and multiple mechanisms and that the inputs and outputs of the mechanism also be multiply realizable. In short, medium-independence allows for the use of physical variables with traits other than voltage (as in typical digital computers); this is imperative in considering other types of computation, such as that occurs in the brain or in a quantum computer. A rule, in this sense, provides a mapping among inputs, outputs, and internal states of the physical computing system. [6]

Giunti calls the models studied by computation theory computational systems, and he argues that all of them are mathematical dynamical systems with discrete time and discrete state space.[7]:ch.1 He maintains that a computational system is a complex object which consists of three parts. First, a mathematical dynamical systems DS{\displaystyle DS} with discrete time and discrete state space; second, a computational setup H=(F,BF){\displaystyle H=\left(F,B_{F}\right)}, which is made up of a theoretical part F{\displaystyle F}, and a real part BF{\displaystyle B_{F}}; third, an interpretation IDS,H{\displaystyle I_{DS,H}}, which links the dynamical system DS{\displaystyle DS} with the setup H{\displaystyle H}.[8]:pp.179–80